Conduction electrons in a magnetic field

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Abstract

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In this paper, three problems that involve the behavior of conduction electrons in a magnetic field are considered: the change of resistance in a magnetic field, the setting up of a tightly bound electron approximation in a magnetic field, and the effect of a magnetic field on Flouquet's theorem.
To treat the change of resistance of monovalent metals in a magnetic field, we use the nearly free electron approximation. Analytical methods for the evaluation of the integrals that arise in the usual treatment are developed by expressing the energy and relaxation time in terms of series of cubically symmetrical spherical harmonics. Numerical results of the correct order of magnitude can be obtained for the Hall coefficient and the coefficients of the change of resistance. However, contrary to the available observations, the ratio of the transverse to the longitudinal change of resistance has a minimum value of about four. The breakdown of Ohm's law at high current densities is considered and a qualitative, physical discussion of the various phenomena is given.
In an attempt to get wave functions that satisfy Schrodinger's equation when the magnetic field is included in the Hamiltonian, we consider the extension of the tightly bound electron approximation in which the atomic wave functions are those for atoms in a magnetic field. The problem is set up, the necessary integrals are evaluated, and the problem is reduced to the solution of a set of simultaneous equations. Various methods of solution are considered in detail, but the general case proves to be intractable. When considering boundary conditions, we find that in order to have cyclic boundary conditions the magnetic field must be limited to a constant that is inversely proportional to the square of the period of the cycle. Some indication of the properties of a general crystal is obtained by determining the energy levels and the magnetic susceptibility of a two by two square lattice.
A theorem fundamental to much of the electron theory of metals is Floquet's theorem, which states that the wave functions of an electron in a crystal lattice can always be chosen so that they have the form [...] where [...] has the periodicity of the lattice. We find that the usual proof of this theorem breaks down in a magnetic field and it appears that the ordinary form of the theorem does not hold there. It would also seem that the important method invented by Wigner and Seitz, and extended by Slater, for the treatment of cohesion is not valid in a magnetic field.